Statistics for Environmental Engineers

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The simulation methods used in Chapter 51 can also be used to investigate the effect of uncertainty in design inputs on design outputs and decisions. They are especially useful when real variability in inputs exists and the variability in output needs to be investigated (Beck, 1987; Brown, 1987).


It is a serious disappointment to learn after an experiment that the variance of computed values is too large. Avoid disappointment by investigating this before running the experiment. Make an analysis of how measurement errors are transmitted into calculated values. This can be done when the model is a simple equation, or when the model is complicated and must be solved by numerical approximation.


Beck, M. B. (1987). “Water Quality Modeling: A Review of the Analysis of Uncertainty,” Water Resour. Res., 23(5), 1393-1441.

Berthouex, P. M. and L. B. Polkowski (1970). “Optimum Waste Treatment Plant Design under Uncertainty,” J. Water Poll. Control Fed., 42(9), 1589-1613.

Box, G. E. P., W. G. Hunter, and J. S. Hunter (1978). Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, New York, Wiley Interscience.

Brown, L. C. (1987). “Uncertainty Analysis in Water Quality Modeling Using QUAL2E,” in Systems Analysis in Water Quality Measurement, (Advances in Water Pollution Control Series), M. B. Beck, Ed., Perga-mon Press, pp. 309-319.

Mandel, J. (1964). The Statistical Analysis of Experimental Data, New York, Interscience Publishers.


49.1    Exponential Model. The model for a treatment process is y = 100 exp(-kt). You wish to estimate k with sufficient precision that the value of y is known within ±5 units. The expected value of k is 0.2. How precisely does k need to be known for t = 5? For t = 15? (Hint: It may help if you draw y as a function of k for several values of t.)

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